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2

For any irreducible representation of over , the matrix is a scalar matrix.

Note first that although is not algebraically closed, remains irreducible over , the algebraic closure. By Step (1), the matrix is a homomorphism of representations, so by Fact (2) (Schur's lemma) it is a scalar matrix with entries over . Of course, since the matrix is defined over , the scalar entries come from .

3

The matrix is the zero matrix for any irreducible representation of over .

By Step (2), the matrix is scalar. Its trace is where is the character of , and this is times , which is zero by assumption.

4

The matrix is the zero matrix for any representation of over .

is splitting -- every representation of over splits completely as a sum of irreducible representations. Note that by Fact (3), the complete reducibility follows from the characteristic not dividing the order of .

For the regular representation, the matrix is a combination of linearly independent permutation matrices with s as coefficients. For this to be zero, all the values must be zero.

Characters form a basis for the space of class functions

Step no.

Assertion/construction

Facts used

Given data used

Previous steps used

Explanation

1

The characters of inequivalent irreducible representations are linearly independent on account of being an orthonormal set. In other words, if for distinct irreducible characters and , then all the equal .

For any irreducible character , taking the inner product with of the right side gives minus a summation where all terms are zero except the term . This simplifies to , so the difference is zero. We thus get that for all irreducible characters .

4

, so for any class function .

Step (3), Previous half of proof which shows that there any class function orthogonal to all irreducible characters is zero

Step-combination direct.

5

The characters form a basis (in fact, an orthonormal basis) for the space of class functions.

Steps (1), (4)

Step (1) shows they are linearly independent, Step (4) shows that they span the space. Together, this means they are a basis.